Onset of a Propagating Self-Sustained Spin Reversal Front in a Magnetic System

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by  P. Subedi

Onset of a Propagating Self-Sustained Spin Reversal Front in a Magnetic System
P. Subedi,1 S. V´elez,2 F. Maci`a,1 S. Li,3 M. P. Sarachik,3
J. Tejada,2 S. Mukherjee,4 G. Christou,4 and A. D. Kent1
1Department of Physics, New York University, New York, New York 10003, USA
2Grup de Magnetisme, Dept. de F´ısica, Universitat de Barcelona, Spain
3Department of Physics, City College of New York, CUNY, New York, New York 10031, USA
4Department of Chemistry, University of Florida, Gainesville, Florida 32611, USA
The energy released in a magnetic material by reversing spins as they relax toward equilibrium
can lead to a dynamical instability that ignites self-sustained rapid relaxation along a deflagration
front that propagates at a constant subsonic speed. Using a trigger heat pulse and transverse
3
andlongitudinalmagneticfields,weinvestigateandcontrolthecrossoverbetweenthermallydriven
1
0 magnetic relaxation and magnetic deflagration in single crystals of Mn12-acetate.
2
PACSnumbers: 75.50.Xx,82.33.Vx,75.40.Gb,75.60.Jk
n
a
J
Raisingthetemperatureofaflammablesubstancecan

0
ignitecombustion,anexothermicreactionbetweenasub-
1
stance and an oxidizer that results in a chemically mod-

] ified substance [1]. Deflagration is a self-sustained com-
ll bustion that propagates at subsonic speed via thermal
a
conduction; locally burning substance increases the tem-
h
perature of adjacent unburnt substance and ignites it.
-
s Deflagration is a dynamic instability of a system gov-
e
m erned by local reactions and diffusion. These systems
FIG.1: a)Thepotentialenergywithdiscreteenergylevelsof
are ubiquitous in nature–from cell growth to epidemics.
. Mn -AcspinHamiltonian,Eq.1. b)Schematicofthesample,
t The study of these nonlinear dynamical systems reveals 12
a theHallsensors,theheater,andthedirectionsoftheapplied
m rich phenomenology–including traveling waves, dissipa- magnetic fields.
tive solitons [2], and self-organized patterns [3, 4].
-
d Magneticrelaxationisadiffusion-reactionprocessthat
n can develop an instability and proceed as a magnetic de- field H = (H ,H ) the simplest spin Hamiltonian has
o ⊥ z
flagration. It can occur in a magnetic material prepared the form:
c
[ in a metastable spin configuration; here the reaction is
2 the reversal of spins that release Zeeman energy and the H=−DSz2−gµBHzSz−gµBH⊥S⊥. (1)
diffusionservestotransmittheenergytoadjacentmate-
v
rial. Thephenomenonhasbeenobservedasfastmagneti- The first term defines the anisotropy barrier (DS2 ≈ 65
0
0 zation jumps in molecular magnets [5, 6], manganites [7] K) and the second and third terms are the Zeeman en-
9 and intermetallic compounds [8]. In these systems, spins ergy corresponding to the field applied parallel and per-
1 that are in a metastable state at low temperature slowly pendicular to the anisotropy axis. As shown in Fig.1a,
.
1 relax to a lower energy state releasing heat. In some the energy barrier between the metastable and the sta-
0 circumstances, the heat cannot be compensated by ther- ble state is the activation energy, U, and the energy
3
maldiffusion,resultinginaninstabilitythatgivesriseto released by the reversing spins is the Zeeman energy,
1
a front of rapidly reversing spins traveling through the ∆E =2gµ H S . The transverse field H mixes eigen-
: B z z ⊥
v sample at constant speed. states of S reducing the activation energy and leading
z
i
X In this Letter we study the instability that leads to to relaxation by quantum tunneling of magnetization,
magnetic deflagration in a thermally driven Mn -Ac particularly at specific resonant values of magnetic field
r 12
a crystal. Weexplorehowtocontrolthecrossoverbetween (H = kD/(gµ ) (cid:39) 0.45k T, for integer k), correspond-
z B
slow magnetic relaxation and rapid, self-sustained mag- ing to the crossings of energy levels with opposite spin-
neticdeflagrationusingmagneticfields. Basedonearlier projections. The longitudinal field H biases the poten-
z
work[9,10], weproposeasimplemodelandperformnu- tial well also reducing the activation energy and increas-
merical simulations that provide a good description of ing the relaxation rate.
the experimental findings. Theabruptcompletereversalofthemagneticmoment
Mn -Ac molecules are magnetically bistable [11] due of Mn -Ac crystals was first reported by Paulsen and
12 12
totheirhighS =10spinandamagneticanisotropythat Park [12] and later shown to take the form of a propa-
providesalargeenergybarrierbetweenspin-upandspin- gating spin reversal front [5]. Magnetic deflagration has
down states (see Fig.1a). In the presence of an applied beenstudiedbysweepingthemagneticfieldorbyraising
2
a.
the crystal’s temperature. Thresholds for both tempera- H = 1.5 T
tguorveer[n13ed]abnydqaupapnltieudmZleaewmsadnufieetldosq[u1a4n]twuemretfuonunnedlitnogboef u.)2 Hz= 0.0 T 1st 3rd 4th 5th 6th
a.
magnetization [15]. In the experiments reported below, (
the spin reversal was triggered using a heat pulse at one B x1
end of the sample.
Magnetic measurements were performed on three 00 20 40 60 80
Time (ms)
Mn -Ac single crystals of dimensions ∼ 0.3×0.3×2.1
mm132, 0.35×0.35×1.75 mm3 and 0.4×0.4×1.6 mm3 b. 3 H = 1.5 T 5th
Hz= 0.6 T 3rd 4th 6th Sensor
(samples A, B and C, respectively). The samples were
mounted on a one-dimensional array of Hall sensors (ac- u.)2 1st
tivearea20×100µm2 with200µmseparation)(seeFig. a.
1b). Thethreesamplesshowessentiallythesamebehav- B (x1
ior; here we show data for sample C. Care was taken to
0
align the sample and the Hall array (placed in the y-z 0.0 0.4 0.8 1.2 1.6
Time (ms)
plane) relative to each other and relative to the magnet
axes, as shown in Fig.1b. The Hall sensors detect the
FIG.2: SignalsoftheHallsensorarraysasafunctionoftime.
stray field B (see inset of Fig.2b), which is maximum
x a)Atzerobiasfield,thepeaksareverybroad,thepropagation
at the reversal front. Measurements were taken at 0.4
time is long and the speed is not constant. b) At H =0.61
z
K in a 3He refrigerator in a 3D vector superconducting T,thepeaksaresharpandspinreversalpropagatesathigher
magnet capable of producing bipolar bias fields up to and constant speed. The inset shows stray fields from the
1 T and bipolar transverse fields up to 5 T. A 6 V, 30 samplewhenthereversalfrontisinthemiddleofthecrystal.
ms pulse was applied to a thin film heater (R ≈ 1.32 kΩ The maximum Bx is at the position of the front.
at 0.4 K) placed at one end of the sample to trigger spin
reversal. A dc current of 20 µA was supplied to the ar-
ray of hall sensors and the signal from each sensor was betweenpulsessensedbyadjacentHallsensorsandtheir
amplified by a factor of 1000, filtered, and recorded by a separation.
data acquisition card. While the curves in Fig.2a and Fig.2b appear simi-
Atbasetemperature(0.4K),thecrystalwasprepared lar, closer inspection shows that the reversal process at
in a fully magnetized state so that all the spins were low and high bias are distinctly different. At zero (or
alignedinonedirection. AfixedH wasappliedandH low) bias, the time interval between adjacent peaks is
⊥ z
wasswepttothedesiredfieldandheld. Atthislowtem- relatively long and increases as the spin-reversal propa-
perature the spins are essentially blocked by the strong gates, indicating that the spin reversal is slowing down;
magnetic anisotropy of Mn -Ac, and relax slowly to- the time to completion is on the order of 80 ms; and the
12
ward equilibrium. In this non-equilibrium state, a volt- peaksarefairlybroadandbroadenfurtherasthereversal
age pulse was then supplied to the heater to increase the proceeds. Here the spin reversal propagates slowly and
temperature at one end of the crystal to initiate spin itsspeeddecreasesbecausethemagnetizationrelaxesto-
reversal. The same pulse was used to trigger the data ward equilibrium following the heat pulse; its progress
acquisition card to record the magnetization signals of through the crystal is governed by thermal diffusivity.
different Hall sensors during 1s. The procedure was re- By contrast, in the case of high bias shown in Fig.2b,
peated for several values of H for a particular H . the time between adjacent peaks is constant and on the
z ⊥
Figure2 shows the time evolution of the Hall sensor order of 100 µs, corresponding to a constant speed of
signals when a heat pulse is applied at t = 0 to sample v ∼2 m/s, consistent with the data measured by Suzuki
C with the magnetization prepared as described above. et al. [5]; the total time for reversal of the full mag-
Two cases are considered: a zero bias field (H = 0 T, netization of the crystal is of the order of 1 ms; and
z
Fig.2a), and a large bias field (H = 0.61 T, Fig.2b) the peaks in Fig.2b are substantially sharper and do not
z
for the same H = 1.5 T. A propagating spin reversal broadenastheytravel. Theconstantspeedinthecaseof
⊥
region is observed in both cases; the change in magne- a large Hz is a signature of magnetic deflagration, a self-
tization always starts at the heater end and propagates sustained process driven by the Zeeman energy released
towards the other end (from 1st to 6th sensor, Fig.1b). by the spins as they reverse.
As shown in Fig.2, a peak in the signal corresponding Wewillnowshowthatthecrossoverbetweenthesetwo
to the enhancement of the local stray field (B ) travels regimes is surprisingly sharp. The speed of propagation
x
in the z-direction as the spin reversal propagates along of the spin reversal fronts was determined from the time
the easy axis, ultimately reversing the magnetization of intervalbetweenmaximaoftheHallsensorsinthemiddle
the entire crystal. The speed of propagation of the re- of the crystal (sensor 3 and 4) and their spatial separa-
versing spins can be determined from the time difference tion. Note that while the same speed is obtained using
3
a . b . HereT isthetemperature,κisthethermaldiffusivity,C
H ^ (T ) 0 .1 0 D e fla g ra tio n is the heat capacity and Γ=Γ exp[−U(h ,h )/(k T)]
0 ⊥ z B
5 0 .0 is the relaxation rate. For fields that are much smaller
1 .0
/s)4 1 .5 /s)0 .0 8 than the anisotropy field (hi =gµBHi/2DS (cid:28)1) [19]:
2 .5
ed (m3 3 .0 ed (m0 .0 6 U(h⊥,hz)≈DS2(1−hz)2(cid:20)1−2h⊥(1(1−−hh2zz)1)/22(cid:21). (3)
e e
p p0 .0 4 T h e rm a l
S2 S R e la x a tio n Forsomeappliedfieldsandinitialconditions,thether-
mal diffusivity cannot compensate for the increase in
1 0 .0 2
temperature due to the Zeeman energy released by the
reversing spins; the sample temperature then rapidly in-
0 0 .0 0
-0 .2 0 .0 0 .2 0 .4 0 .6 -0 .2 0 .0 0 .2 creases and a magnetic deflagration develops. To study
H z ( T ) H z ( T ) the ignition of magnetic deflagration, we consider a sys-
tem of spins blocked in a given metastable magnetic
FIG. 3: a) The speed of propagation of the spin reversal as state, increase its temperature (e.g., from 0.5 to 5K),
a function of H for different H . b) A blow-up of the data
z ⊥ and study the resulting nucleation process. Let us con-
forsmallH . Thespeedissmallandnearlyconstantatvery
z sider here for simplicity the temperature evolution in a
smallH andincreasesabruptlywhenadeflagrationdevelops.
z
nucleation volume, independent of coordinates,
any pair of Hall sensors for large Hz, the speed varies T˙ =m˙ ∆E/C−2κ/R2(T −T0) (4)
from point to point at low H , and therefore between
z
different Hall sensor pairs. As shown in Fig.3, the speed where T is the temperature of the volume under study,
of the spin reversal front changes abruptly at Hz that T0 isthetemperatureoftherestofthesampleand2R is
depends on the transverse field. For a given H , below thecharacteristicsizeofthenucleationvolume. Theneg-
⊥
a crossover bias field H (for e.g., H = 0.25 T for H ative term in Eq. 4 is linear with temperature while the
co co ⊥
= 0 T), the speed of propagation of the reversing spins reaction term increases exponentially with temperature
is nearly independent of the bias field. However, above (m˙ ∝ exp[−U/(kBT)]); there is thus a competition be-
the crossover, the speed increases suddenly and depends tween these two terms and stationary solutions (T˙ = 0)
strongly on the bias field. Figure3b, a magnification of correspond to temperatures when the dissipation equals
Fig.3a where the vertical scale has been expanded by a the released energy.
factor of 100, demonstrates this even more clearly. As shown in the inset of Fig.4, some magnetic field
The peaks in the speed at Hz ∼ 0.45 T shown in configurations (H = (H⊥,Hz)) corresponding to small
Fig.3a are a clear manifestation of the important influ- bias fields have stationary solutions (red dotted curve).
ence of quantum mechanics on the dynamics of the sys- As we vary H (i.e., we vary the relaxation rate, m˙),
tem. These maxima are due to quantum tunneling at the stationary points move closer to each other and
theresonantfields[15,16]wheneverspinstatesonoppo- eventually merge (green dashed curve) and disappear.
site sides of the anisotropy barrier have nearly the same When there is no stationary solution (blue solid line),
energy. As noted earlier, tunneling enhances relaxation the temperature derivative is always positive leading to
and effectively reduces the anisotropy barrier. As we in- aninstability–amagneticdeflagrationmaydevelop. The
creaseH⊥weincreasethetunnelsplittingbetweenlevels, condition for the crystal to lose stability at given T0 and
which results in a broadening of the resonance steps. A H is T˙ >0 and its crossover points are found by solving
small peak can also be seen in Fig.3b at the zero bias
resonance. Note that the zero-field resonance is slightly T˙ =0, ∂T˙/∂T =0. (5)
shifted (∼40 mT) due to internal dipolar fields [17, 18].
To better understand the crossover between the two AtagivenH⊥,forHz <Hco,theenergyreleaseddur-
regimes described above, we develop a simple model ing the spin reversing process is small (or even zero in
based on previous theory [9, 10]. The diffusion-reaction the case of zero bias). The dynamics of the propagating
system that describes the time evolution of the magne- front observed in this case (Fig. 2a) are determined by
tization, m, towards equilibrium, m , entails two pro- thediffusionoftheheatsuppliedbytheheaterpulse;this
eq
cesses: the Zeeman energy (i.e. heat) released into the correspondstoathermalregime. Herethereactionterm
samplebythespinsastheyreverseandtheheatdiffusion in Eq. 4 is negligible and the equation has a stable solu-
through the sample. The dynamical system of nonlinear tion at a temperature close to the sample temperature.
partial differential equations reads: NeglectingthereactionterminthefullsetofEqs.2,itis
clearthattheheatdiffusesandthemagnetizationlocally
m˙ =−Γ(m−m ) T˙ =m˙ ∆E/C+∇·κ∇T. (2) follows the sample temperature.
eq
4
oftheMn -Accrystalisawellcontrolledprocess. Once
M e ta s ta b le (H ^ = 1 T , H z = 0 .1 T ) 12
p o p u la tio n , n 4 (1 , 0 .0 6 ) thecrystalisintheZeemanregimethespeedofthemag-
1 .0 ) netic deflagration can be controlled by the strength of
0 .2 0 .8 . K/s2 (1 , 0 .0 5 ) theapplied field. Additionally, equalpropagationspeeds
0 .7 (
) T0 are obtained for different combinations of applied fields,
T
( as seen in Fig.3 where a particularly speed can be seen
z -2
to correspond to different magnetic field configurations.
H 4 .5 5 .0 5 .5
0 .1 T ( K ) Larger bias fields would produce larger amounts of heat
D e fla g r a tio n released and higher temperatures for the overall process;
T h e r m a l
while, large transverse fields that speed up the process
R e la x a tio n
by the same amount would heat the sample much less.
0 .0
0 1 2 3 Wehavemeasuredthedynamicsofspinreversalfronts
H ^ ( T )
in Mn -Ac crystals in the presence of both bias and
12
transversemagneticfields. Wehaveshownhowtheseap-
FIG.4: Mainpanel: Theboundarycurvesseparatingthermal
pliedfieldscanbeusedtocontrolaninstabilitythatsepa-
and Zeeman regimes for various n. The blue dots are H ,
derivedfromthedatainFig.3. TheinsetplotsT˙ asafuncticoon rates slow magnetic relaxation from rapid, self-sustained
magnetic deflagration.
of T for three different H. The solid and open black circles
indicatestableandunstablesolutionsforthe(1,0.05)Tfield We have also expanded the range of conditions under
case. which magnetic deflagration has been observed, particu-
larlytothecaseofsmallbiasfields,whichcorrespondsto
smallenergyreleaseinthedeflagrationprocess. Thisisa
For Hz ≥ Hco, the energy released by the reversing particularly interesting limit in which instabilities in the
spins is large, the reaction term dominates over the dif- front are predicted to occur, such as pattern formation
fusiontermandEq.4displaysaninstability. InthisZee- andoscillationsofthefrontpositionasitpropagates[22].
man regime, the released energy drives the spin relax- Inthepresenceoflargetransversefields,wherequantum
ation to a deflagration with steady propagation, where tunnelingdominatesoverthermaleffects,theinstabilities
the speed of the propagating front of reversing spins in the magnetization dynamics may result in supersonic
strongly depends on both Hz and H⊥. fronts [23] and lead to a deflagration-to-detonation tran-
The sharp transition in propagation speed from ther- sition[24,25]. Thepresentexperimentsthusprovideop-
maltoZeemanregimesasafunctionoftheappliedmag- portunities to study–in a nondestructive manner–a large
netic field is given by the condition in Eqs.5. H⊥ varies variety of instabilities in magnetic systems.
the activation barrier and does not change the released We acknowledge illuminating discussions with Dov
energy; it affects only m˙ in the reaction term. Hz affects Levine. ADK and PS acknowledges support by NSF-
both m˙ and ∆E. DMR-1006575andNYU.FMacknowledgessupportfrom
InthemainpanelofFig.4,weplottheboundariesthat a Marie Curie IOF 253214. MPS acknowledges support
separatethethermalregimefromtheZeemanregimede- from NSF-DMR-0451605. Support for GC was provided
rived from Eq.5 together with the experimental points under grant CHE-0910472. SV acknowledges financial
(blue dots) obtained from Fig.3. The different curves supportfromtheMinisteriodeEducaci´on,CulturayDe-
denote different initial metastable spin population (n = porte de Espan˜a and JT acknowledges financial support
(Ms−m)/(2Ms),whereMs issaturationmagnetization) from ICREA Academia.
that determines the initial state [26]. For fields above
the boundary curve, the reaction term dominates and
the crystal develops a magnetic deflagration; below the
curves, thermal diffusion dominates and the spins evolve
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